1 Introduction

Co-simulation is commonly used for connecting sub-models with independent numerical solvers. This technique may improve modularity and reusability of simulation models, facilitate collaboration between organizations, and enable the use of the most suitable modeling environment and solver algorithm for each part of a larger model. However, numerical instability remains a key issue in co-simulation, especially when connecting stiff mechanical sub-models. Sampled communication between isolated solvers may introduce time delays on interface variables. These may cause instability or inaccuracy if left unhandled. Furthermore, aliasing effects may arise if the sampling rate is too low compared to the time scale of the sub-models [1,2,3]. A well-known solution to these problems is to use Transmission Line Modeling (TLM) [4,5,6]. TLM addresses the stability problems by ensuring that the introduced time delays are physically motivated. As a consequence, no non-physical delays need to be introduced, and numerical errors can be completely avoided.

While TLM offers the possibility to effectively eliminate numerical instability, it comes at the cost of enforcing a fixed global maximum communication step-size that is limited by the physical time delay in the elements being modeled. Such a fixed global maximum step-size can negatively impact the simulation performance if a quantified coupling error is unacceptable for accuracy or stability reasons. In most cases, the internal solver integration step-size in a co-simulation unit is not allowed to be larger than the communication step-size. Thus, the steps of the internal solver will also be limited by the physical delays. This can severely limit simulation performance, especially if the time scale of a sub-model is significantly larger than the timescale enforced by the TLM connection.

In this paper, we propose the use of distributed adaptive communication delay-size control. When transients are small, it should be possible to relax the maximum step-size limitation and temporarily allow larger delays. Steady-state identification is suggested for controlling delay-size. The purpose of the presented research is to enable numerically stable co-simulation without compromising simulation performance.

1.1 Research questions

The following topics are investigated in the paper. Example models are used for numerical assessments.

  1. 1.

    Bi-lateral delay lines with asymmetric delays Traditional TLM connections, also known as bi-lateral delay lines, have symmetric numerical properties. As a consequence, the delays have the same magnitude in both directions. However, the proposed approach to distributed delay-size control requires asymmetric delays. It must be ensured that this is feasible and that it does not compromise simulation accuracy.

  2. 2.

    Matching characteristic impedance The physical properties of a TLM connection are computed from the combination of the delay and the characteristic impedance. The characteristic impedance \(Z_c\) is the ratio between input pressure and input flow of an infinitely long transmission line, see for example Boden et al. [7], and it quantifies the resistance a transmission line presents to changes in pressure. It must be investigated whether or not it is necessary to change the impedance accordingly to match the change in the delays.

  3. 3.

    Varying delays and impedance during simulation For the proposed method to be effective, the delays and characteristic impedance of the TLM connection must change online during co-simulation. This must be possible without introducing non-physical numerical delays.

  4. 4.

    Error criteria for delay-size control Controlling the size of the delays requires some criteria. Alternatives will be identified and their feasibility investigated.

  5. 5.

    Evaluation of performance improvements Finally, if the method is feasible, a numerical assessment of the effects on stability and performance must be conducted.

1.2 Related work

An alternative solution to the proposal presented herein is to invoke an adaptive communication step-size [8]. In such a case, the communication step-size for the entire composite model is modified to meet specified tolerances. Adaptive communication step-size can be implemented either using implicit communication schemes [8, 9], explicit schemes [10], or both [11]. In comparison to distributed adaptive delay-size, as described in this paper, an implicit adaptive communication step-size control requires all sub-models to internally support so called solver rollback, that is the capability of the solver to revert to a previously stored stage in the simulation. Hence, the co-simulation will be limited to include only sub-models supporting this feature. Furthermore, with adaptive communication step-size, the communication step will be reduced everywhere, even for parts of the composite model that might not need it, possibly rendering sub-optimal execution times.

Another related technique is dynamic decoupling [12, 13]. With dynamic decoupling, models are split into weakly coupled sub-models by identifying nodes with large capacitance, where effort variables change slowly, or large inductance, where flow variables change slowly. This minimizes the error from the communication delay. A drawback is that models can only be decoupled in nodes with slow dynamics, and the transient behaviors of the interface variables are not considered.

A third related approach is to make communication between weakly coupled sub-models more accurate through the use of extrapolation of interchanged variables. Accuracy can be further improved by analyzing the current trend of each variable and adopting the extrapolation method accordingly. Both synchronous approaches [14, 15] and asynchronous algorithms using explicit error estimation [16] exists. Éguillon et al. [10] later presented F\(_3\)ORNITS, a co-simulation algorithm providing flexible extrapolation, time-stepping, and scheduling. In comparison with the method proposed in this paper, all of the aforementioned methods rely on weak coupling methods and cannot on their own guarantee numerical stability for a general case model.

Other researchers have previously investigated TLM with adaptive delay-size control. One approach is to use parasitic inductance for error control when simulating fluid systems [17]. There are also similar studies for thermal diffusion [18] and electrical circuits [19]. These studies focused on parallel simulation in a single simulation tool and did not consider co-simulation.

2 Background

2.1 Steady-state identification

SSI techniques can be useful in many different applications ranging from automated model calibration and validation [20] to simulation step-size control. Krus exploited SSI for managing evolving model timescales for optimal simulation performance in [21]. In [22] Hallqvist et al. applied and evaluated four different well-established SSI techniques on aircraft in-situ measurements. The evaluation criteria focused on objectivity and scalability in addition to the SSI functionality. These criteria are also the most relevant for the research presented in this article.

Three methods for online SSI are analyzed here. The first is a rectangular sliding window (RSW) with an absolute tolerance. When the difference between the maximum and minimum value of a signal y(t) of interest in the window is below the specified tolerance \(y_{\text {tol}}\), then the signal is considered to be at steady state

$$\begin{aligned} \begin{gathered} {\text {SS}}(t_i) = \max \left( y(t)\right) -\min \left( y(t)\right) < y_{\text {tol}}\\ \forall t \in [ t_i-\Delta t_{\text {w}},t_i ], \end{gathered} \end{aligned}$$
(1)

where the length of the sliding window is represented by \(\Delta t_{\text {w}}\) in Eq. (1). This method requires two parameters to be specified, the window length and the absolute tolerance. Both of these need to be calibrated depending on both the order of magnitude of the signal and the expected oscillation frequency. On the other hand, the required implementation effort is small and the method is easy to understand, which are characteristics that should not be underestimated particularly when considering technology adoption in industry.

A more sophisticated approach is to apply the F-test type statistic known as the ratio of differently estimated variances [23], which operates on a sliding window of data just as the technique presented in Eq. (1). The ratio

$$\begin{aligned} R=\frac{\sigma _1^2}{\sigma _2^2} \end{aligned}$$
(2)

is equal to one at steady state if there is no random noise present. In the case of SSI for step-size control, there is no random noise enforcing a subjective threshold value on R. In Eq. (2), the variance is estimated in two different ways: as the mean square deviations from the average \(\mu\)

$$\begin{aligned} \sigma _1^2(y) =\frac{1}{n-1} \sum _{i=1}^n(y_i-\mu )^2 \end{aligned}$$
(3)

and as the mean of squared difference of successive data [24]

$$\begin{aligned} \sigma _2^2(y) =\frac{1}{n-1} \sum _{i=1}^{n-1}(y_{i+1}-y_i)^2. \end{aligned}$$
(4)

The current data set in the sliding window is denoted by y in both Eqs. (3) and (4). Furthermore, \(\mu\) is the average value of all data in the current window, continuously estimated across the n samples in the sliding window.

Cao et al. proposed an improvement to the F-test method by exploiting an exponentially weighted moving average instead of a sliding window when estimating the two variances of Eq. (2) [23]. This improvement, denoted as R-test, reduces the overhead associated with the method, but it does introduce a total of three subjective parameters, \([\lambda _1, \lambda _2, \lambda _3]\)that need to be specified. The first variance is computed as the mean square deviation from the filtered data as

$$\begin{aligned} \sigma _1^2(y)= \frac{2-\lambda _1}{2}v_{f,i}^2 = \frac{2-\lambda _1}{2}\lambda _2\left( y_i-y_{f,i-1}\right) ^2 v_{f,i-1}^2, \end{aligned}$$
(5)

where

$$\begin{aligned} y_{f,i} = \lambda _1 y_i+\left( 1-\lambda _1\right) y_{f,i-1} \end{aligned}$$
(6)

is the filtered data. The second variance is computed using the mean square differences of successive filtered data, \(\delta _{f,i}^2\), as

$$\begin{aligned} \sigma _2^2(y)= \frac{\delta _{f,i}^2}{2}=\frac{\lambda _3\left( y_i-y_{i-1}\right) ^2+\left( 1-\lambda _3\right) \delta _{f,i-1}^2}{2}. \end{aligned}$$
(7)

If the simulation is assumed to start at steady state, the initial value for \(y_{f,0}\) becomes the initial value for the signal and \(\delta _{f,0}^2\) becomes zero.

Even though both the F-test and R-test approaches to SSI are developed to be general to any input signal, generality issues arise when applying the techniques on simulation results without noise. Periodic oscillations always generate a non-steady condition if noise is absent. This means that there is no generic tunable parameter to separate irrelevant oscillation amplitudes from relevant ones without modifying the techniques. Measured signals always include appended noise, and the magnitude of this noise is typically related to the magnitude of the measured quantity via the range of the sensor; a relationship that introduces a generic calibration parameter via a tolerance on Eq. (2). One way to mitigate this drawback on noiseless modeled signals could be to include modeled noise; however, the noise connection to the range of the quantity of interest also needs to be modeled for the solution to be general. This method is used in the following experiments.

2.2 Transmission line modeling

The TLM boundary equations can be derived from the wave equation and the telegrapher’s equation [25]. They describe the relationship between flow variables (f) and effort variables (e) on each side of a TLM element (Fig. 1).

Fig. 1
figure 1

A TLM element defines a relationship between flow and effort variables on each side

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The effort variables on either side of the element

$$\begin{aligned} \begin{aligned} e_1(t) = e_2(t-\Delta t) + Z_c\big [f_1(t)+f_2(t-\Delta t)\big ]\\ e_2(t) = e_1(t-\Delta t) + Z_c\big [f_2(t)+f_1(t-\Delta t)\big ] \end{aligned} \end{aligned}$$
(8)

depend on the flow variable on the same side and the flow and effort variables on the other side. \(Z_c\) is the characteristic impedance of the element. Note that the TLM element is symmetric; see Eq. (8).

The two main properties of the element are the time delay \(\Delta t\) and the characteristic impedance \(Z_{\text {c}}\). These together yield the capacitance

$$\begin{aligned} C = \frac{\Delta t}{Z_{\text {c}}} \end{aligned}$$
(9)

and inductance

$$\begin{aligned} L = Z_{\text {c}}\Delta t \end{aligned}$$
(10)

of the element where C and L define the physical behavior depending on the element’s domain. In a mechanical element, for example, the capacitance and inductance correspond to the compressibility and inertia, respectively.

The TLM equations enable asynchronous co-simulation [4]. Two interconnected sub-models interact by sending flow variables to the TLM element whenever they are produced. In the TLM element, a co-simulation master algorithm is used to store the flow variables in interpolation tables. Whenever a sub-model requires an effort variable, it can then be interpolated for a particular simulation time. A prerequisite for this to work is that each sub-model must be able to produce output variables at a frequency equal to or larger than the inverse of the TLM time delay. In practice, when working with integrating solvers for systems of differential equations, this means that the internal integration step in the sub-models must be equal to or smaller than \(\Delta t\). If the communicating sub-models are unable to produce data simultaneously, then the integration step must be limited to \(\Delta t/ 2\) [26].

The time delay in a TLM connection corresponds to an inherent physical property of the modeled element. Hence, no numerical time delays are required. TLM thereby offers co-simulation with guaranteed numerical stability as analytically shown by Braun et al. in [27]. TLM with asynchronous communication using interpolation tables is currently implemented in the OMSimulator master simulation tool [28], which was used for the experiments in this paper.

3 Delay-size control

The limitation on the internal integration steps in the sub-model solvers may induce performance limitations. If the limitation on the integration step can be relaxed, it may be possible to achieve numerical stability without compromising performance. The implications of such a relaxation on the physical properties of the TLM connection are investigated analytically in this section.

As mentioned, the two main properties in a TLM element are inductance and capacitance. The relationships between L, C, e, and f are

$$\begin{aligned} e_L(t) = L\frac{{\text {d}}f_L}{{\text {d}}t} \end{aligned}$$
(11)

and

$$\begin{aligned} f_{\text {C}}(t) = C\frac{{\text {d}}e_{\text {C}}}{{\text {d}}t}, \end{aligned}$$
(12)

where the subscripts indicate the origin of the contribution to the variable in question. Hence, L produces an effort proportional to the change rate of the flow, while C produces a flow proportional to the change rate of the effort. At steady state, it holds that

$$\begin{aligned} \frac{{\text {d}}f}{{\text {d}}t} \approx 0 \end{aligned}$$
(13)

which combined with Eq. (11) shows that the effects of both L and \(e_L\) are negligible in steady state.

We now introduce a new factor \(\kappa\), which is the increase factor for the delay. Both the characteristic impedance and the time delay are then multiplied by \(\kappa\)

$$\begin{aligned} Z'_{\text {c}} = \kappa Z_{\text {c}},\end{aligned}$$
(14)
$$\begin{aligned} \Delta t' = \kappa \Delta t, \end{aligned}$$
(15)

where \(1 \le \kappa \le \kappa _{\text {max}}\). Furthermore, \(Z'_{\text {c}}\) and \(\Delta t'\) render a new inductance

$$\begin{aligned} L' = Z'_{\text {c}}\Delta t' = \kappa Z_{\text {c}}\kappa \Delta t = \kappa ^2 L \end{aligned}$$
(16)

and a new capacitance \(C'\)

$$\begin{aligned} C' = \frac{\Delta t'}{Z'_{\text {c}}} = \frac{\kappa \Delta t}{\kappa Z_{\text {c}}} = C. \end{aligned}$$
(17)

As can be seen, \(C' = C\) while L increases by the square of \(\kappa\). This means that when close to steady state, it should be possible to increase the TLM time delay if it can be compensated for by increasing the characteristic impedance by the same magnitude. The capacitance will not change, and the inductance will not affect the results at steady state.

3.1 Asymmetric communication delays

Because the described delay-size control is conducted within each included sub-model, each sub-model computes its own delay and the corresponding compensated characteristic impedance. As a result, the proposed method must support asymmetric delays, thus enabling delay-size control in a distributed fashion rather than letting the master algorithm control the delays. Additionally, support for asymmetric delays enables sub-models that support delay-size control to be connected to sub-models that do not. This provides a flexibility and modularity similar to the work presented in [10].

3.2 SSI for delay-size control

A TLM connection is here considered to be at steady state if both the effort and flow variables are at steady state intuitively rendering a multivariate SSI problem. As highlighted by Xu et al. [29], multivariate SSI is often dealt with by means of combining all relevant univariate conditions via a logical and to identify process overall steady state. Even though more advanced multivariate SSI techniques exist, for example, Xu et al. who expand on the R-test by means of forming a ratio between differently estimated co-variance matrices instead of the variances [29], the traditional approach where

$$\begin{aligned} {\text {SS}}_{\text {process}}=\prod _{i=a}^{b} SS(i) \ \end{aligned}$$
(18)

was used for the purposes of this article.

The obvious approach to identifying steady state in a TLM connection is to equate Eq. (18) for \(i=e_{\text {L}}\) and \(i=f_{\text {C}}\), as described in Eqs. (11) and (12), exploiting a suitable univariate SSI technique. However, as described earlier in this section, analyzing the physics of the TLM connections renders a second univariate option via Eq. (13).

An SSI algorithm is used here to determine whether the delay-size should be increased or decreased. Figure 2 shows how delay-size control is implemented within a sub-model. After each finished internal solver step, the algorithm checks for steady state. Should steady state not be detected, and the delay is not at the minimum, the delay will be reduced and the solver will roll back and restart with a shorter step. How much the delay is reduced or increased at every step may likely affect simulation performance, but this is not further investigated in this work. In the experiments presented below, a simple method is used where the delay is either halved or doubled until it reaches the maximum or minimum size. This strategy is here deemed as sufficient for the purposes of this prototype implementation; however, future research on the topic could prove valuable for performance improvements. The internal solver contains a callback function for evaluating model state derivatives. Input variables are automatically fetched from within this function, and does therefore not appear in the flowchart.

Fig. 2
figure 2

Flowchart describing the implementation of distributed delay-size control

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4 Test models

A total of three test models, consisting of two or more sub-system models coupled using TLM, serve as the foundation for evaluating the technology developed herein.

4.1 Spring-mass model

The first model consists of a 1D spring-mass system with three masses; see Fig. 3. It is divided into two sub-models, where the rightmost contains two of the masses. Thus, the sub-model contains the second-order internal dynamics. After 0.1 s, the leftmost mass is subjected to a step force. The CVODE [30] solver is used to solve the system of ordinary differential equations

$$\begin{aligned} {\left\{ \begin{array}{ll} M\ddot{x_1} + B\dot{x_1} = F_{\text {step}} - F_{\text {tlm}},\\ M\ddot{x_2} + B\dot{x_2} + K_1(x_2-x_3) = F_{\text {tlm}},\\ M\ddot{x_3} + B\dot{x_3} + K_1(x_3-x_2) + K_2x_3 = 0. \end{array}\right. } \end{aligned}$$
(19)

Furthermore, the model parameters are listed in Table 1.

Table 1 Parameters used in the spring-mass model
Full size table
Fig. 3
figure 3

A test model consisting of a three-mass system divided into two sub-models. The rightmost sub-model has second-order internal dynamics

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As reference, the model is simulated without delay-size control.

4.2 Drive cycle model

The second test model is an electric vehicle simulated performing a drive cycle. While the model is quite simple, it still represents a more realistic scenario compared to the first test model. The model is separated into a sub-model of a DC motor and a vehicle sub-model; see Fig. 4.

Fig. 4
figure 4

A model of an electric car powered by a DC motor performing a drive cycle

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The relationship between torque (\(T_{\text {tlm}}\)), velocity (\(\omega _{\text {tlm}}\)), voltage (V), and current (I) in the motor can be expressed as

$$\begin{aligned} V = IR + \frac{{\text {d}}I}{{\text {d}}t}L + K_{\text {e}} \omega _{\text {tlm}} \end{aligned}$$
(20)

and

$$\begin{aligned} T_{\text {tlm}} + \frac{{\text {d}}\omega _{\text {tlm}}}{{\text {d}}t}J + B\omega _{\text {tlm}} = K_{\text {t}} I, \end{aligned}$$
(21)

where the parameters for Eqs. (20) and (21) are provided in Table 2.

Table 2 Parameters for the electric car test model
Full size table

The motion of the car is simulated by applying Newton’s second law with rolling resistance, air drag, and gravitational pull. Again, the CVODE solver is used to solve the system of ordinary differential equations. The car is simulated performing the New European Drive Cycle (NEDC) together with some variations in road altitude.

4.3 Hydraulic robot model

The third and final model used for the evaluation is a 3-dimensional (3D) hydraulic robot. This model serves two different validation purposes in the presented context. First, it validates the method by incorporating an adaptive delay-size on a 3D mechanical problem. Second, because the model contains connections between sub-models that support adaptive delay-size and those that do not, it is also used to validate the flexibility of the method.

A schematic overview of the robot model is provided in Fig. 5. It performs a three-stage motion

  1. 1.

    Raise the arm by approximately \(25^{\circ }\)

  2. 2.

    Rotate \(90^{\circ }\)

  3. 3.

    Lower the arm.

Fig. 5
figure 5

A hydraulic robot model is connected to two models of electro-hydraulic actuators. The latter models support delay-size control

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The motion is controlled by two electro-hydraulic actuators (EHAs), as shown in Fig. 6. For the rotating EHA, the hydraulic piston is replaced by a hydraulic motor. An EHA is a self-contained actuator with an electric motor, a hydraulic pump, a hydraulic actuator, and possibly control valves. Such an actuator eliminates the need for a central hydraulic supply system and makes it possible for each actuator to operate independently. Accordingly, the two EHAs are modeled as independent sub-models. The 3D mechanical system is modeled using the Modelica standard library [31, 32] and OpenModelica [33]. The latter sub-model does not support adaptive communication delays, whereas the two EHA sub-models do.

Fig. 6
figure 6

Flow-controlled electro-hydraulic actuator (EHA) used in the 3D model

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5 Experiments

5.1 SSI for delay-size control

The hypothesis that the SSI of a TLM connection is a univariate problem, under the assumption presented in Eq. (13), is tested by comparing univariate SSI to the results from traditional multivariate SSI, here realized by invoking Eq. (18).

5.2 Adaptive delay-size

The suggested method was validated by allowing the maximum delay-size for both interconnected models to increase 20 times when steady state was identified. Each model controlled its own delay-size by identifying steady-state conditions using the RSW method. This relaxed the maximum limitation on the solver integration step-size and thereby the number of derivative evaluations. The delay-size and the number of derivative evaluations were logged to verify the usefulness of the method.

5.3 Asymmetric communication delays

An important property of the proposed method is that it must support asymmetric delays. This makes it possible for each sub-model to control the delay-size in a distributed fashion rather than letting the master algorithm control the delays. In this way, sub-models supporting delay-size control can be connected to sub-models that do not.

Asymmetric delays were tested by simulating the spring-mass model with \(\kappa _1=1\) and \(\kappa _2=10\). Hence, the second sub-model was allowed to increase the delay-size ten times when steady state was observed, while the first sub-model was forced to always use the physically motivated delay. In this way, it could be verified that a sub-model supporting delay-size control can be connected to a regular model using TLM.

6 Results

The results from applying the experiments described in Sect. 5 on the test models, see Sect. 4, are presented in this section.

6.1 SSI for delay-size control

Simulated effort and flow variables from all three test models served as the foundation for comparing the different SSI techniques. The SSI results related to the spring-mass model are shown in Fig. 7, the drive cycle model in Fig. 8, and the hydraulic robot model in Fig. 9.

The RSW univariate and multivariate approaches rendered close to identical steady-state intervals in all investigated experiments, indicating that a univariate approach is sufficient for TLM connections. In other words, the effort variable does not have an impact on the results if the flow variable is in steady state. Several different advantages related to generality emerge as a result. All of the investigated univariate SSI methods have drawbacks in terms of generality within the presented context and need to be calibrated to the range of the input signal. The univariate solution renders half the number of calibration parameters compared to multivariate in the case of SSI on a TLM connection.

Fig. 7
figure 7

Results from SSI on spring-mass velocity using the three univariate methods and the multivariate RSW

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Fig. 8
figure 8

Results from SSI on angular velocity in the drive cycle model using the three univariate methods and the multivariate RSW

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Fig. 9
figure 9

Results from SSI on piston and motor velocities for the robot model using the three univariate methods and the multivariate RSW

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No further evaluation concerning the selection of which univariate method to employ, in combination with the assumption of Fig. 13, was done here. However, results from all methods are presented to demonstrate that they all perform similarly if calibrated to do so. Modeled noise was appended to the input signal of the F-test and R-test methods to allow for calibration.

6.2 Adaptive delay-size

Simulation results for the spring-mass model when using adaptive delay-size with \(\kappa _1 =\kappa _2 =20\) are shown in Fig. 10 along with the corresponding results generated without an adaptive delay-size. No significant differences can be observed when comparing the two solutions.

Fig. 10
figure 10

Velocity in the spring-mass model with and without delay-size control

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Figure 11 shows the delay-size used by each sub-module when \(\kappa=20\). Both models now use a larger delay-size at the beginning and end of the simulation, where steady-state conditions are identified. Note that the delay-sizes are roughly similar but also contain noticeable differences.

Fig. 11
figure 11

Communication delays for the spring-mass sub-models with \(\kappa _1=\kappa _2=20\)

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In Fig. 12, the number of derivative evaluations with and without adaptive delay-size is shown. The black line indicates the number of evaluations with the reference simulation without delay-size control (\(\kappa _1=\kappa _2=1\)). The workload for the simulation was reduced from approximately 7000 to 4300 evaluations for each sub-model, a reduction of 38%.

Fig. 12
figure 12

Total number of derivative evaluations for the spring-mass sub-models with \(\kappa _1=\kappa _2=20\). The number of derivative evaluations when a fixed step-size is applied is shown as solid, whereas the evaluations with an implemented variable steps size is shown as dashed

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Fig. 13
figure 13

Vehicle speed in the drive cycle model with and without delay-size control

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A similar experiment was performed on the drive cycle model. Vehicle speed is shown in Fig. 13, while Fig. 14 shows the delay-size used by each sub-module. Delay-size was increased to the maximum when the car was moving at constant speed. The number of derivative evaluations for the two sub-models is shown in Fig. 15. The black line indicates the number of evaluations with the reference simulation without delay-size control (\(\kappa _1=\kappa _2=1\)). In total, the simulation requires approximately 58% fewer evaluations with delay-size control compared to without.

Fig. 14
figure 14

Communication delay for the car and motor sub-models with \(\kappa _1=\kappa _2=20\)

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Fig. 15
figure 15

Number of derivative evaluations for the car and motor sub-models with \(\kappa _1=\kappa _2=20\)

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Simulation results and communication delay for the robot model are shown in Figs. 16 and 17, respectively. As expected, both EHAs use maximum delays when standing still and a smaller delay when moving.

Figure 18 shows the number of derivative evaluation for the two EHA sub-models of the robot model. The number of evaluations in the mechanical 3D model is not of interest, because it does not support adaptive delay-size control. The black reference line indicates the number of derivative evaluations without delay-size control (\(\kappa _1=\kappa _2=1\)). The two solvers need approximately 44% fewer evaluations with delay-size control.

Fig. 16
figure 16

Simulation results from the robot model with and without delay-size control

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Fig. 17
figure 17

Communication delays for the two EHA sub-models with \(\kappa _1=\kappa _2=20\)

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Fig. 18
figure 18

Number of derivative evaluations for the two EHA sub-models with \(\kappa _1=\kappa _2=20\)

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6.3 Asymmetric delays

The simulation results from running the spring-mass model with delay-size control only in sub-model 2 are shown in Fig. 19. No issues regarding accuracy or stability can be observed.

Fig. 19
figure 19

Velocity in the spring-mass model with and without asymmetric delay-size control

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Figure 20 shows the delay-size used by sub-model 2. As before, the delay-size is increased at the beginning and end of the simulation where the model is found to be at steady state. This confirms that sub-models supporting delay-size control can be connected with TLM to those that do not.

Fig. 20
figure 20

Communication delay for sub-model 2 with \(\kappa _1=1\) and \(\kappa _2=10\)

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7 Conclusions

We show that it is possible in practice to vary the time delays in a TLM connection. To preserve the dynamic properties of the coupling, the characteristic impedance must be adjusted to compensate for the varying time delay. Furthermore, we also demonstrate that asymmetric TLM connections, with different delays in different directions, are feasible. This enables distributed adaptive delay-size control, where each sub-model decides its delay-size by monitoring its state variables to identify steady-state conditions. It also makes it possible to connect sub-models that support delay-size control with those that do not.

Three SSI algorithms were investigated, and all three provided satisfying criteria for delay-size control. In addition, univariate SSI on the flow variable of a TLM connection was analytically and experimentally shown to be sufficient for identifying steady state. This reduces the number of SSI parameters that need to be calibrated by a factor of two regardless of which of the presented traditional SSI methods is employed.

A significant reduction in the number of derivative evaluations was found when comparing simulations with an adaptive step-size to those with a fixed step-size. This is a promising result, because derivative evaluation is usually the most time-consuming task in continuous-time simulation.

The proposed method depends on the tuning of several parameters. This includes parameters for SSI, which are specific for each SSI method, but also the maximum allowable delay increase and the minimum time threshold between two periods of steady state. Unfortunately, some of these parameters are dependent on model properties and thus cannot easily be generalized. This includes, for example, the amplitude tolerance in the RSW method and the standard deviation of the white noise in the F-test and R-test methods. However, because the delay-size control is handled by each sub-model, the parameters can be chosen by the modeler.

A benefit of letting each sub-model independently handle the delay-size control is that the solution becomes independent of the master simulation tool. Any master algorithm that supports interpolation tables for delayed variables will work with the proposed method. This also includes tools supporting the Functional Mock-up Interface (FMI). FMI version 3.0 supports intermediate update mode, where the importing tool and the sub-model can exchange variables at any time between communication points [34]. This enables asynchronous TLM co-simulation [27]. Even though intermediate update is not available in FMI 2.0, the behavior can be mimicked by letting each sub-model handle the interpolation of variables [6]. The methods presented in this paper are compatible with both versions of the standard.